Squares and square roots

Classroom resources

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Students will be able to

• I can work out the square of a number and list the perfect squares to 144.
• I can find the square root of a perfect square and explain what it means.
• I can find the side and the perimeter of a square from its area.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need times tables and area as side × side (Year 6).

The one idea: squaring is a number times itself (the area of a square), and the square root goes back (the side). The common mistakes are squaring as doubling (5² = 10) and rooting as halving (√36 = 18); both are planted.

Recall of the perfect squares to 144 is the hinge for the next lesson (roots between whole numbers).

Running the lesson

1. Predict: take a show of hands. Students who chose 10 doubled.
2. Run: ask how the board knew the area. Count one row, then the rows.
3. Investigate: write 6² = 36 and √36 = 6 on the board as a pair that undo each other.
4. Modify: before the perimeter task, ask what the 49 is: an area or a length?
5. Make: collect squares past 144 and their roots; look at the last digits.

The hard part (on purpose)

Telling squaring from doubling under time pressure. Ask students to draw the square for 5² before they answer, rather than correcting the 10.

When students get stuck

Answers to the checks

1. 5 squared: What is $5^2$?
   25. 5² means 5 × 5: a square with side 5 has 5 rows of 5 unit squares. 10 is 5 × 2, double, not squared.
2. a square of 49: Change the side $s$ until the square's area is 49. What is $\sqrt{49}$? Was your prediction for $5^2$ right?
   
3. Undo the square: Work out $\sqrt{81}$.
   9
   9 × 9 = 81, so √81 = 9.
4. Find the mistake: Ava finds the area of a square with side 5 cm. Click the line where her working goes wrong.
   line 3
   Line 3: 5² means 5 × 5 = 25, not 5 × 2. The area is 25 cm².
5. Why it works: Why is $\sqrt{64} = 8$?
   Because $8 \times 8 = 64$
   The square root of 64 is the number that times itself makes 64: 8 × 8 = 64.
6. The biggest in the list: Make a square with area 144 m².
   
   12 × 12 = 144, so √144 = 12.
7. A root: Work out $\sqrt{121}$.
   11
   11 × 11 = 121.
8. A side from an area: A square tile has area 64 cm². How long is each side, in cm?
   8
   √64 = 8, so each side is 8 cm.
9. Two steps: the perimeter: A square garden has area 49 m². What is its perimeter, in metres?
   28
   Side = √49 = 7 m. Perimeter = 4 × 7 = 28 m.
10. Not a perfect square: Edge case: is 50 a perfect square?
    No: 7² = 49 and 8² = 64, and no whole number squares to 50
    50 is between 49 (7²) and 64 (8²), so no whole number squares to 50. The next lesson finds roots like √50.
11. Square: Work out $9^2$.
    81
    9 × 9 = 81.
12. Root: Work out $\sqrt{100}$.
    10
    10 × 10 = 100.
13. A side: A square rug has area 9 m². How long is each side, in metres?
    3
    √9 = 3, so each side is 3 m.
14. Review: integers: Review: work out $-4 \times 3$.
    -12
    4 × 3 = 12, and the signs differ, so −12.
15. a bigger square: Make a perfect square bigger than 144 (past the list).
    
    13² = 169, 14² = 196, 15² = 225: so √169 = 13, √196 = 14, √225 = 15.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › The biggest in the list

Open task: any program that does what the step asks.

one change at a time › A root

Open task: any program that does what the step asks.

one change at a time › A side from an area

Open task: any program that does what the step asks.

one change at a time › Two steps: the perimeter

Open task: any program that does what the step asks.

one change at a time › Not a perfect square

Open task: any program that does what the step asks.

a bigger square

Open task: any program that does what the step asks.

Stretch and support

Stretch: The last digits of square numbers are 0, 1, 4, 5, 6 or 9. Use this to show that 2023 and 1458 cannot be perfect squares.

Support: Keep the list of squares to 144 visible, and use the grid board for every root.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Squares and square roots".

The ideas:
- Squares as area models: I can connect squaring a number to the area of a square with that side length, and list perfect square numbers from 1 to 144.
- Square root as inverse operation: I can Students understand that square root is the inverse of squaring and can find the exact square root of any perfect square up to 144.

What understanding looks like:
- Lists perfect square numbers from 1² to 12² without error.
- Represents a given perfect square as an area model and states its side length.
- Does not confuse the square number with its root (e.g. does not say '9 squared is 3').
- States the square root of any perfect square from 1 to 144 when prompted.
- Explains in words what √25 means ('the number that when multiplied by itself gives 25').
- Does not conflate squaring with multiplying by 2.

Mistakes students often make here:
- A student writes 5² = 10 (Ava's mistake).
- A student halves for a root (√36 = 18).
- A student gives the side when the question asks for a perimeter.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Squares and square roots".

Exit ticket

A square has area 81 cm². Find its side and its perimeter.

Side √81 = 9 cm; perimeter 4 × 9 = 36 cm.

What each step is for

• 5 squared: Committing to a value for 5² shows whether they read the small 2 as 'times 2' (10) or as 'times itself' (25).
• a square of 49: Growing the side until the area is 49 shows the square root as the side of a square with that area.
• squaring and its inverse: Squaring multiplies a number by itself, and the square root undoes it; Ava's mistake (5² = 10) is squaring as doubling, and its mirror is halving for a root.
• one change at a time: One change at a time: the biggest square in the list, a root, a side from an area in context, a perimeter (two steps), then a number that is not a perfect square.
• squares and roots: Squares, roots and a side from an area, mixed; doubling is the planted slip; the review keeps integers fresh.
• a bigger square: Choosing a perfect square past 100 and naming its root extends the list beyond 12² by the same reasoning.

Where it leads

Next: Between the squares (nodes 3 and 4), then exponent notation (AC9M7N02).

Where it leads: Pythagoras (Year 8) uses squares and roots on every side of a right-angled triangle.